APCOM'07 in conjunction with EPMESC XI, December 3-6, 2007 ...

ζ
−1
ζ
imp
( ∇ × F ) s ( ) s n ( ∇ × E)
l − F ( ) ( ) L ( ) s kˆ
2
, ˆ µ −
, s−n
El
= − jω
Fs
, Jˆ
2
Ω
2
s 2
2 D
L ( Ω ) L ( Ω )
n=
s+
2
∑
n=
s−2
2 D
2 D
2. 2D Stokes problem We consider the SUPG (Streamline Upwind Petrov-Galerkin [14]) stabilized weak
formulation of the Stokes problem: Find velocity and pressure fields ( , p ) ∈ ( u + V)
× Q where
1
V = { v ∈ H ( Ω)
: v = 0 on ΓD
} and = ( Ω)
2
L
∫ µ & ε ( ) : & ε ( v)
Ω − ∫ p∇
o vdΩ
= ∫ ρb
o vdΩ
+ ∫
Ω
−
Ω
Ω
Q such that (9-10):
ΓN
u D
2 u d t o vdΓ
(9)
∫
Ω
∑∫
K∈Th K
∑∫
K∈Th K
N
( ∇ o s(
) )
q∇ o vdΩ − τ ∇q
o ∇pdK
= τ ∇q
o ˆ u dK . (10)
h
h
The SUPG formulation is utilized to solve the plane flow of an isothermal fluid in a square lid-driven
0 , 1 × 0,
1 presented in Fig. 5. Fluid dynamic viscosity is defined as µ = 1 and the body force
cavity ( ) ( )
b = 0 . The stabilization coefficient is defined as τ
K
2
αhK
= with = 0.
01
2µ
α .
3. Non-stationary heat transfer problem The weak form of the non-stationary heat transfer problem:
Find the temperature distribution u uD
+ V V =
1
v ∈ H Ω : v = 0 on Γ satisfying
∈ where ( )
{ }
( ρc pu
v)
+ ∫ k∇u
o ∇v
dΩ
+ ∫ βuv
dΓN
= ∫ fv dΩ
+ ∫ ( βu
N + q)
v dΓN
∀v
∈V
Ω
&, (11)
Ω
ΓN
( u(
) , v)
= ( ρc
u , v)
∀v
∈V
c p 0
Ω p 0
Ω
Ω
ΓN
ρ (12)
FE - discretization in time gives the following matrix system:
M u&
+ Ku = f
(13)
Applying the trapezoidal rule for the time discretization we obtain
k + 1
k k
( M + αδ K)
u = [ M − ( 1−
α ) δ K]
u + δ f
where M is the mass matrix, δ is the time step, ∈[
0,
1]
α gives different time integration schemes. We
focus on the solution of the heat-transfer problem in the L-shape domain presented in Fig. 6.
Fig. 6 Geometry of the step problem
The initial temperature distribution is 0 0 = u at 0 = t . The L-shape domain is heated/cooled with 1 ± = u N
with β = 1 and no internal heating f = 0 .
D
∀F
s
(8)
(14)